Supremum of increasing net of operators on Hilbert space

Question

Consider the following fragment of Murphy's "$C^*$-algebras and operator theory".

Can someone explain how $\sup_\lambda u_\lambda$ is defined (is it just notation?) and why it exists?

Relevant definitions: $B(H)_{sa}$ are the self-adjoint operators in $B(H)$ and a net $(u_\lambda)$ in $B(H)$ converges strongly if and only if $\Vert u_\lambda(x) - u(x)\Vert \to 0$ for all $x \in H$.

Answer 1

"Sup" means "least upper bound". As the net is increasing, $u_\lambda\leq u$ for all $\lambda$; so $u$ is an upper bound.

If $u_\lambda\leq v$ for all $\lambda$, then $$ \langle ux,x\rangle=\lim_\lambda\langle u_\lambda x,x\rangle\leq\langle vx,x\rangle. $$ So $u$ is below any upper bound for the net. Thus, $u$ is the least upper bound.